On integrability of transverse Lie-Poisson structures at nilpotent elements (Q778975)

Let \(Q = L_1 + \mathfrak{g}^f\) be the Slodowy slice corresponding to an \(sl_2\)-triple \(\{L_1, h,f\}\) in a simple Lie algebra \(\mathfrak{g}\) of rank \(r,\) where \(L_1\) and \(f\) are nilpotent elements and \(h\) is semisimple. The author considers the case where \(L_1\) belongs to a certain list of distinguished nilpotent orbits of semisimple type and to the corresponding transverse structure \(B^Q\) on \(Q\) of the Lie-Poisson structure on \(\mathfrak{g}.\) By applying the argument shift method, the main result shows that sufficiently many polynomial functions can be produced to give completely integrable systems on \(Q\) equipped with the transverse structure.